World!Of
Numbers

WON plate
179 |


[ February 2, 2011 ]
[ Last Update January 21, 2013 ]
All numbers above 23 are either Powers or Sums Of Powers.

Combining ninedigital (and pandigital) squares (and cubes) with their
reversals and expressing them as sum of powers.
- goto second part

Our Beloved Prof V.K Doraswamy - A Memoir - goto memoir

by B.S. Rangaswamy (email)

On the topic Sum Of Powers (WON plate 178), I had stated that
" With the liberty to use squares, cubes...
All numbers above 23 are either powers or sums of powers
".
Enclosed is a sheet to substantiate this statement, which may
be of interest to you and other Math lovers.

It is possible to express EVERY NUMBER above 23,
as Sum Of Powers !
To arrive at the least number of constituent
powers is an intellectual task.
Rudiments of this exercise are depicted below.

Evolved readers can think of devising suitable Java script
to instantly arrive at the various constituent powers
for high value numbers.

Numbers
> 23
Sum Of Powers Numbers
> 23
Sum Of Powers
2424 + 23 7472 + 52
2552 7552 + 52 + 52
2632 + 32 + 23 7662 + 62 + 22
2733 7733 + 52 + 52
2824 + 23 + 22 7872 + 52 + 22
2952 + 22 7933 + 33 + 52
3032 + 32 + 23 + 22 8026 + 24
3133 + 22 8134
3225 8226 + 32 + 32
3324 + 32 + 23 8372 + 52 + 32
3424 + 32 + 32 8426 + 24 + 22
3533 + 23 8572 + 62
3662 8625 + 33 + 33
3752 + 23 + 22 8762 + 33 + 24 + 23
3852 + 32 + 22 8826 + 24 + 23
3933 + 23 + 22 8972 + 25 + 23
4025 + 23 9034 + 32
4152 + 24 9126 + 33
4252 + 32 + 23 9272 + 33 + 24
4333 + 24 9334 + 23 + 22
4462 + 23 9472 + 62 + 32
4562 + 32 9526 + 33 + 22
4652 + 32 + 23 + 22 9626 + 25
4733 + 24 + 22 9734 + 24
4825 + 24 9872 + 72
4972 9972 + 52 + 52
5052 + 52 100102
5133 + 24 + 23 10134 + 24 + 22
5233 + 52 10272 + 72 + 22
5372 + 22 10372 + 33 + 33
5433 + 33 104102 + 22
5533 + 24 + 23 + 22 10526 + 25 + 32
5625 + 24 + 23 10672 + 72 + 23
5725 + 24 + 32 10726 + 33 + 24
5872 + 32 108102 + 23
5925 + 33 109102 + 32
6062 + 24 + 23 11034 + 52 + 22
6172 + 23 + 22 11126 + 33 + 24 + 22
  OR
72 + 72 + 32 + 22
6272 + 32 + 22 112102 + 23 + 22
6362 + 33 11334 + 25
6426 11472 + 72 + 24
6572 + 24 11534 + 52 + 32
6672 + 32 + 23 116102 + 24
6772 + 32 + 32 11734 + 62
6826 + 22 118102 + 32 + 32
6972 + 24 + 22 11934 + 52 + 32 + 22
7033 + 33 + 24 120102 + 24 + 22
7133 + 62 + 23 121112
7226 + 23 12234 + 25 + 32
7372 + 24 + 23 12372 + 72 + 52


From B.S. Rangaswamy [ November 19, 2011 ]

A few days back, I came across a moving vehicle bearing reg. no '9441'.
Last number 441 cajoled me to set out in quest of the constituent
powers of this 4 digiter :

Vehicle NumberSum Of Powers# tiers
9441212 + 942 + 122 + 24 + 225 tier
212 + 942 + 102 + 264 tier
212 + 902 + 3023 tier
212 + 203 + 1033 tier
213 + 122 + 623 tier
962 + 1522 tier
972 + 252 tier

Thus it was possible to arrive also at 2 tier results. This is to illustrate
that all numbers have several sets of such constituent powers. To arrive
at the least number of constituent powers is really an intellectual task.


[ November 2011 ]
Combining ninedigital (and pandigital) squares (and cubes) with their
reversals and expressing them as sum of powers.

by B.S. Rangaswamy

With due regards to Mr Albert H Beiler, I have attempted to find
typical sets of constituent powers related to each
of ninedigital & pandigital squares, depicted in your webpage
The Nine Digits Page 2 under the following headings.

A. Reverse of ninedigital squares (30 nos)
B. Reverse of pandigital squares (87 nos)
C. Palindromic twin ninedigitals (30 nos)
D. Palindromic twin pandigitals (87 nos)

No pandigital cubes exist in 10 digit numbers. Only 6 cubes with
all numerals from 0 to 9 exist in eleven digit series !
(Episode 24 'Cubes with all numerals' of my book "Wonders of Numerals" )

E. Reverse of elevendigital cubes (6 nos)
F. Palindromic twin elevendigitals (12 nos)
G. Palindromic 32 digitals (6 nos)
H. Palindromic 63 digitals (10 nos)

Enclosing detailed sums of powers for all the above combinations,
for your kind perusal and considerations for inclusion
in your renowned website World!Of Numbers.


A000179 Prime Curios! Prime Puzzle
Wikipedia 179 Le nombre 179



A. Reverse Ninedigital Squares as Sum of Powers
Sl.NoPower 
Ninedigital Square
Reversal of
Ninedigital Square
Sum Of Powers
1234392549386721127683945112982 + 1502 + 1292
2271292735982641146289537120952 + 29
3196292385297641146792583121132 + 373 + 1312
4230192529874361163478925127812 + 3422 + 203
5156812245893761167398542129382 + 732 + 372
6195692382945761167549283129412 + 2812 + 292
7264092697435281182534796135022 + 4342 + 2062
8244412597362481184263795135712 + 2752 + 1272
9250592627953481184359726135772 + 1462 + 592
10259412672935481184539276455 + 65 + 153
11255722653927184481729356219462 + 3142 + 622
121807223265971844817956237833 + 11282 + 783
13231782537219684486912735220662 + 153 + 103 + 22
14291062847159236632951748251582 + 1532 + 153
15290342842973156651379248255162 + 5242 + 144
16303842923187456654781329255872 + 2942 + 182
17203162412739856658937214256672 + 3772 + 142
18242762589324176671423985259112 + 1802 + 1082
19118262139854276672458931259312 + 2032 + 312
20146762215384976679483512260582 + 6422 + 2282
21193772375468129921864573303622 + 1022 + 55
22190232361874529925478163304172 + 5252 + 932
23272732743816529925618347304232 + 2372 + 572
24248072615387249942783516306962 + 803 + 303 + 102
25125432157326849948623751307992 + 2012 + 702 + 72
26242372587432169961234785310002 + 224 + 232
27159632254817369963718452310342 + 7642 + 1602
28123632152843769967348251311012 + 2732 + 392
292288725238147699674183259893 + 66 + 104
30267332714653289982356417313422 + 1872 + 222


B. Reverse Pandigital Squares as Sum of Powers
Sl.NoPower 
Pandigital Square
Reversal of
Pandigital Square
Sum Of Powers
163051239754286011068245793326842 + 412 + 28
280361264578903211230987546350852 + 1392 + 104 + 103
387639276805943211234950867351412 + 2452 + 312
462961239640875211257804693354632 + 3902 + 1682
562679239286570411407568293375132 + 5802 + 822
657321232856970411407965823375222 + 2552 + 172 + 52
789079279350682411428605397377932 + 5422 + 282
863129239852706411460725893382172 + 4202 + 1022
972621252738096411469083725383262 + 3402 + 2932
1097779295607328411482370659384972 + 5692 + 1672
1155581230892475611657429803407092 + 4512 + 612
1283919270423985611658932407407292 + 2792 + 552 + 102
1368781247308259611695280374411732 + 2512 + 382
1475759257394260811806249375424972 + 5042 + 152 + 53
    or
715 + 11002 + 304 + 24 + 23
1598802297618352044025381679634412 + 7832 + 902 + 32
1690198281356792044029765318634802 + 2332 + 54 + 22
1791248283261975044057916238637002 + 4672 + 902 + 72
1843902219273856044065837291637632 + 3412 + 292
1996702293512768044086721539639272 + 2272 + 912 + 202
2084648271652839044093825617639832 + 64 + 25
2148852223865179044097156832640082 + 3482 + 1082
2256532231958670244207685913648632 + 6622 + 1702 + 104
2376182258036971244217963085649432 + 5442 + 2702 + 103
2454918230159867244276895103653972 + 3552 + 372 + 102
2578072260952371844817325906694062 + 3532 + 902 + 192
2681222265970132844823107956694482 + 2442 + 1542
2735172212370695844859607321697102 + 3112 + 1602 + 302
2838772215032679844897623051699832 + 74 + 192
2989145279468310255201386497721172 + 6182 + 3782
3066105243698710255201789634721222 + 4452 + 902 + 54
3194695289671430255203417698721332 + 4972 + 103
3298055296147830255203874169721372 + 3502 + 702
3381945267149830255203894176721382 + 542 + 63
3489355279843160255206134897721512 + 5642 + 2002 + 104
3591605283914760255206741938721562 + 5012 + 512
3658455234169870255207896143721652 + 3302 + 32 + 32
3740545216438970255207983461721652 + 583 + 210 + 102
3880445264713980255208931746721722 + 3292 + 1612
3937905214367890255209876341721792 + 2302 + 1202 + 103
4061575237914806255260841973725292 + 6042 + 1462
4177346259824037166173042895785652 + 7492 + 1502 + 132
4295154290542837166173824509785732 + 3182 + 842
4358554234285709166190758243786792 + 5212 + 3192
4455446230742589166198524703787272 + 7152 + 2702 + 72
4550706225710984366348901752796782 + 683 + 602 + 62
4642744218270495366359407281797442 + 234 + 1482
4790144281259407366370495218798132 + 703 + 1932
4833144210985247366374258901798382 + 3562 + 1612
4944016219374082566528047391807952 + 4452 + 1302 + 212
5060984237190482566528409173807982 + 2872 + 104
5199066298140723566532704189808252 + 1502 + 103 + 26
5261866238274019566591047283811852 + 2032 + 432
5365634243078219566591287034925 + 5112 + 4592
5466276243925081766718052934819632 + 3422 + 512
5545624220815493766739451802820912 + 6802 + 2392
5655524230829145766754192803821832 + 3832 + 54
5753976229134085766758043192822062 + 603 + 36 + 33
5855626230942518766781524903823502 + 37 + 63
5937176213820549766794502831824252 + 853 + 203 + 34
    or
18943 + 311 + 105 + 222 + 63
6049314224318705966950781342833702 + 4712 + 512
6192214285034217966971243058834912 + 7042 + 192
6232286210423857966975832401835212 + 2682 + 562
6339336215473208966980237451835472 + 3692 + 34
6478453261458732099023785416949922 + 4242 + 503 + 242
6576047257831462099026413875950072 + 2852 + 512
6690153281275634099043657218950972 + 3602 + 2972
6759403235287164099046178253951102 + 4932 + 1522
6885353272851346099064315827952052 + 4892 + 2912
6935853212854376099067345821952222 + 3412 + 28
7039147215324876099067842351952232 + 5812 + 2902 + 312
7149353224357186099068175342952262 + 4292 + 152
7285803273621548099084512637953122 + 3472 + 1222
7388623278540361299216304587960012 + 3352 + 192
7486073274085613299231658047960812 + 2692 + 303 + 53
7567677245801763299236710854961072 + 3942 + 132
7689523280143675299257634108962102 + 11002 + 2382 + 582
7785743273518620499402681537969672 + 2722 + 922
7835757212785630499403658721969722 + 2972 + 123
7932043210267538499483576201973822 + 234 + 2062
8068763247283501699610538274980332 + 2632 + 24
8169513248320571699617502384980682 + 4122 + 24
8235337212487035699653078421982502 + 1202 + 392
8346587221703485699658430712982762 + 5062 + 502
8458413234120785699658702143982772 + 5752 + 502 + 172
8565637243082157699675128034983612 + 3972 + 2902 + 22
8645567220763514899841536702992032 + 4822 + 2632
8771433251026734899843762015992152 + 3712 + 902 + 72


C. Palindromic Twin Ninedigitals as Sum of Powers
Sl.NoPowerPalindromes - 18 digits
Ninedgt. Sq._Reversed Ninedgt. Sq.
Sum Of Powers
1118262139854276_6724589313739709552 + 385852 + 2092 + 103
2123632152843769_9673482513909523882 + 168652 + 1292 + 292
3125432157326849_9486237513966444872 + 296832 + 1732 + 1422
4146762215384976_6794835124640958702 + 113242 + 4402 + 62
5156812245893761_1673985424958767602 + 77002 + 213 + 412
6159632254817369_9637184525047943832 + 292452 + 3032 + 772
7180722326597184_4817956235714868192 + 138222 + 932 + 232
8190232361874529_9254781636015600802 + 87102 + 742 + 37
9193772375468129_9218645736127545422 + 343542 + 1672 + 982
10195692382945761_1675492836188261152 + 236972 + 852 + 210
11196292385297641_1467925836207234822 + 63692 + 1372 + 732
12203162412739856_6589372146424483292 + 349862 + 1572 + 27
13228872523814769_9674183257237504882 + 329702 + 3912 + 802
14230192529874361_1634789257279246942 + 320702 + 172 + 102
15231782537219684_4869127357329527152 + 453952 + 5662 + 1232
16234392549386721_1276839457412062602 + 355592 + 1702 + 1582
17242372587432169_9612347857664412372 + 136802 + 462 + 102
18242762589324176_6714239857676745252 + 183682 + 2062 + 1502
19244412597362481_1842637957728922832 + 79562 + 1172 + 34
20248072615387249_9427835167844662192 + 345052 + 1732 + 512
21250592627953481_1843597267924351582 + 393682 + 1832 + 432
22255722653927184_4817293568086576432 + 299872 + 4032 + 1772
23259412672935481_1845392768203264472 + 392012 + 3152 + 2792
24264092697435281_1825347968351259072 + 252972 + 1532 + 232
25267332714653289_9823564178453716872 + 283042 + 1162 + 242
26271292735982641_1462895378578943062 + 296462 + 4272 + 28
27272732743816529_9256183478624479852 + 556312 + 4752 + 1562
28290342842973156-6513792489181356962 + 195402 + 4042 + 24
29291062847159236_6329517489204125362 + 143822 + 3682 + 1522
30303842923187456_6547813299608264442 + 341322 + 3002 + 1132


D. Palindromic Twin Pandigitals as Sum of Powers
Sl.NoPowerPalindromes - 20 digits
Reversed Pandgt. Sq._Pandgt. Sq.
Sum Of Powers
16305121068245793_397542860132684029632 + 736742 + 5302 + 3302 + 342
28036121230987546_645789032135085432112 + 548062 + 422 + 202
38763921234950867_768059432135141867732 + 463942 + 1662 + 104
46296121257804693_396408752135465542332 + 797082 + 7082 + 522
56267921407568293_392865704137517573122 + 760452 + 5342 + 2542
65732121407965823_328569704137522870672 + 104982 + 1222 + 1082
78907921428605397_793506824137796896652 + 1192762 + 6562 + 2482
86312921460725893_398527064138219443902 + 1171872 + 622 + 123
97262121469083725_527380964138328628012 + 631202 + 3262 + 582
109777921482370659_956073284138501566982 + 201612 + 462 + 402
115558121657429803_308924756140711543852 + 811442 + 3582 + 2062
128391921658932407_704239856140729993952 + 732942 + 1102 + 104
136878121695280374_473082596141173782612 + 239842 + 4502 + 782 + 103
147575921806249375_573942608142499992652 + 569102 + 2662 + 2002 + 303
159880224025381679_976183520463445895682 + 1157002 + 9542 + 922
169019824029765318_813567920463480432562 + 899362 + 3362 + 262
179124824057916238_832619750463701775782 + 1143302 + 7482 + 703 + 24
184390224065837291_192738560463763918412 + 442252 + 4072 + 72
199670224086721539_935127680463927470932 + 655052 + 5112 + 204 + 32
208464824093825617_716528390463983010382 + 655222 + 4262 + 3002 + 502
214885224097156832_238651790464009037102 + 1330182 + 9822 + 342
225653224207685913_319586702464866677982 + 1076322 + 7862 + 703 + 3002
237618224217963085_580369712464945847322 + 1211442 + 5402 + 1582
245491824276895103_301598672465397974762 + 769402 + 4922 + 104 + 222
257807224817325906_609523718469406958632 + 585652 + 2492 + 332 + 102
268122224823107956_659701328469448599382 + 903262 + 8582 + 105 + 303
273517224859607321_123706958469710883782 + 1932622 + 3662 + 403 + 102
283877224897623051_150326798469983019722 + 1421402 + 5402 + 105 + 203
298914525201386497_794683102572120638502 + 383402 + 2302 + 52
306610525201789634_436987102572123433322 + 756612 + 6362 + 214 + 103
319469525203417698_896714302572134719092 + 838742 + 3822 + 2122
329805525203874169_961478302572137883042 + 684652 + 3602 + 282
338194525203894176_671498302572138021712 + 662622 + 1242 + 422
348935525206134897_798431602572153550832 + 648862 + 463 + 2982
359160525206741938_839147602572157757302 + 522902 + 4452 + 103
365845525207896143_341698702572165754642 + 762242 + 2472 + 1122
374054525207983461_164389702572166359622 + 601012 + 3182 + 28
388044525208931746_647139802572172929452 + 1119122 + 5702 + 662
393790525209876341_143678902572179473122 + 1123732 + 10062 + 2462
406157525260841973_379148062572531661862 + 1098072 + 3382 + 502 + 62
417734626173042895_598240371678568714482 + 747282 + 7122 + 782
429515426173824509_905428371678573688402 + 1060702 + 5802 + 3962
435855426190758243_342857091678681371642 + 437102 + 2042 + 482
445544626198524703_307425891678730710002 + 2490642 + 6162 + 582
455070626348901752_257109843679679995932 + 923172 + 6632 + 403 + 36
464274426359407281_182704953679745891922 + 1751102 + 5662 + 153 + 292
479014426370495218_812594073679815382082 + 1495362 + 220 + 1602
483314426374258901_109852473679838956022 + 1657182 + 7982 + 7002 + 22
494401626528047391_193740825680796332782 + 705142 + 184 + 2302 + 302
506098426528409173_371904825680798571602 + 878382 + 583 + 2702 + 202
519906626532704189_981407235680825145772 + 1113392 + 4552 + 592
526186626591047283_382740195681185265182 + 1017742 + 4662 + 1802 + 103
536563426591287034_430782195681186741742 + 270762 + 503 + 702 + 22
546627626718052934_439250817681963729382 + 749242 + 3882 + 23 + 22
554562426739451802_208154937682094164222 + 1796602 + 823 + 182
565552426754192803_308291457682183896242 + 1456422 + 4942 + 603 + 103
575397626758043192_291340857682207318362 + 590702 + 3742 + 2702 + 22
585562626781524903_309425187682350014582 + 1408242 + 8442 + 1202 + 202
593717626794502831_138205497682428774282 + 1354622 + 583 + 1062
604931426950781342_243187059683371346052 + 233072 + 3412 + 1792
619221426971243058_850342179683493970182 + 1559342 + 8462 + 103 + 202
623228626975832401_104238579683521448732 + 1773732 + 10072 + 672
633933626980237451_154732089683547815342 + 1751562 + 4982 + 2302 + 502
647845329023784516_615487320994993602502 + 829122 + 6172 + 1742
657604729026413875_578314620995007441152 + 1306742 + 5642 + 29 + 102
669015329043657218_812756340995098145192 + 484142 + 5662 + 503 + 64
675940329046178253_352871640995111399172 + 1130362 + 743 + 703 + 403
688535329064315827_728513460995206700542 + 123582 + 2272
693585329067345821_128543760995222611922 + 511482 + 383 + 632
703914729067842351_153248760995225219092 + 643922 + 106 + 3922
714935329068175342_243571860995226967512 + 1045422 + 903 + 622
728580329084512637_736215480995312709732 + 1288782 + 4142 + 2702 + 702
738862329216304587_785403612996001586382 + 555652 + 7422 + 64
748607329231658047_740856132996081517722 + 601792 + 5482 + 203 + 103
756767729236710854_458017632996107808492 + 1306682 + 7982 + 5002 + 502
768952329257634108_801436752996216599962 + 968892 + 4062 + 662
778574329402681537_735186204996967425132 + 1180242 + 8002 + 482
783575729403658721_127856304996972463722 + 1095962 + 3402 + 432
793204329483576201_102675384997383654682 + 1506382 + 2102 + 2092
806876329610538274_472835016998033352862 + 1224552 + 6482 + 382
816951329617502384_483205716998068865522 + 321472 + 4162 + 702 + 302
823533729653078421_124870356998250081022 + 829642 + 1872 + 302
834658729658430712_217034856998277315342 + 1333862 + 7242 + 792
845841329658702143_341207856998278696282 + 919072 + 5562 + 1202
856563729675128034_430821576998362228692 + 1251762 + 6962 + 105 + 962
864556729841536702_207635148999204519562 + 1037882 + 3402 + 2472
877143329843762015_510267348999215734712 + 1210422 + 5282 + 4902 + 103


E. Reverse Elevendigital Cubes as Sum of Powers
Sl.NoPowerElevendigit cube
Reversed
elevendigit cube
Sum Of Powers
12535316290480375573084092612393912 + 5742 + 1702 + 22
22795321834609875578906438122406042 + 5642 + 2002 + 302
33506343095878216612878590342475622 + 8732 + 3902 + 312
42326312584301976679103485212605952 + 6002 + 4802 + 212
53123330459021867768120954032771472 + 12762 + 872 + 72
63909359730618429924816037953041072 + 6352 + 503 + 203 + 112


F. Palindromic Twin Elevendigitals as Sum of Powers
Sl.NoPowerPalindrome - 22 digitsSum Of Powers
Elevendigit cube_Reversed cube
12326312584301976_67910348521354743597212 + 2302242 + 343 + 1402 + 402
22535316290480375_57308409261403614672372 + 1840762 + 1462
32795321834609875_57890643812467275185252 + 2316732 + 2872 + 832
43123330459021867_76812095403551896927582 + 2331832 + 3692 + 332 + 102
53506343095878216_61287859034656474509912 + 2134012 + 9952 + 102 + 33
63909359730618429_92481603795772855862562 + 2426212 + 7272 + 303 + 332
Sl.NoPower 
Reversed cube_Elevendigit cube
Sum Of Powers
12535357308409261_16290480375757023178382 + 2659252 + 11202 + 3552 + 34
22795357890643812_21834609875760859013302 + 1527702 + 553 + 402 + 102
33506361287859034_43095878216782865627252 + 3843642 + 155 + 922 + 28
42326367910348521_12584301976824077353902 + 657702 + 3962 + 122 + 24
53123376812095403_30459021867876425098922 + 4014232 + 7652 + 204 + 72
63909392481603795_59730618429961673561012 + 3212282 + 8402 + 7502 + 122


G. Palindromic 32_digitals as Sum of Powers
Sl.NoPowerPalindrome - 32 digitsSum Of Powers
Reversed cube_Repdigital_Elevendigit cube
12535357308409261_9999999999_1629048037575702317839019962 + 577531372 + 111462 + 3452 + 572
22795357890643812_7777777777_2183460987576085901330521002 + 706884752 + 47302 + 352 + 53
33506361287859034_3333333333_4309587821678286562725881212 + 322227772 + 52802 + 57 + 1392
52326367910348521_9999999999_1258430197682407735390556632 + 558919192 + 127422 + 2512 + 912
43123376812095403_5555555555_3045902186787642509893062482 + 2061483 + 836812 + 8072 + 3312
63909392481603795_1111111111_5973061842996167356101283712 + 1381200942 + 162722 + 217 + 1642


H. Palindromic 63_digitals as Sum of Powers
Mid No.Palindrome - 63 digits
Sum Of Powers
ZERO
 
3433683820_29251248465_7849089281_Z_1829809487_5648421529_20283863343
9032 + 13527045085918952 + 449337802 + 62502 + 1132 + 932
1
 
3433683820_29251248465_7849089281_1_1829809487_5648421529_20283863343
9032 + 34394490092985592 + 562003122 + 39342 + 1112 + 292
2
 
3433683820_29251248465_7849089281_2_1829809487_5648421529_20283863343
9032 + 46722381668280612 + 466934932 + 76272 + 3622 + 202
3
 
3433683820_29251248465_7849089281_3_1829809487_5648421529_20283863343
9032 + 56417913367621842 + 1052666312 + 104542 + 1912 + 36
5
 
3433683820_29251248465_7849089281_4_1829809487_5648421529_20283863343
9032 + 64675968866005272 + 1097553722 + 111462 + 2952 + 832
4
 
3433683820_29251248465_7849089281_5_1829809487_5648421529_20283863343
9032 + 71992922907439202 + 548781672 + 3773 + 1852 + 142
6
 
3433683820_29251248465_7849089281_6_1829809487_5648421529_20283863343
9032 + 78631933390680942 + 256410582 + 77272 + 672 + 53
7
 
3433683820_29251248465_7849089281_7_1829809487_5648421529_20283863343
9032 + 84752468688271752 + 898452012 + 110692 + 3502 + 842
8
 
3433683820_29251248465_7849089281_8_1829809487_5648421529_20283863343
9032 + 90459830581073302 + 81042252 + 46042 + 174 + 592
9
 
3433683820_29251248465_7849089281_9_1829809487_5648421529_20283863343
= {Sum Of Powers} ?
Power columns against the last 63 digital palindrome are left blank, inviting enthusiastic
readers to compute appropriate powers. There can be several such choices !
It is an intellectual and educating exercise to weave the last few powers. I remember to have
laid a well designed trap to get a square and cube totalling 8055299079 Vide WON 178.



Our Beloved Prof V.K Doraswamy - A Memoir †


It was a shocking news to learn about the sad demise
of our beloved Prof VKD on 6th November 2012. He retired
as Professor of Mathematics from J.C. College of Engineering
Mysore. VKD tried to popularise Maths in Schools and
colleges for decades. His Kannada book Vinoda Ganita is a
gem and narrates in detail puzzles in Leelavati, a Sanskrit
compendium on Maths and puzzles in Astronomy by
Bhaskaracharya, the great Indian Mathematician. VKD has
authored many text books in Maths for Schools and Colleges.

Prof VKD served The Indian Institute of World Culture
Bangalore as active Member / Vice President for decades. He
was an adept in Sanskrit, Law, Literature and Recreational
Maths.

Lately I got very close with professor, when he presided
over a session on "Palindromes" conducted by me at the
Esteemed Institute in the Palindromic year '2002'. He was
kind enough to write a valuable Foreword to my maiden
book Wonders of Numerals, in the year 2004.

I have met Prof V.K.Doraswamy on several occasions after
that and the last one was at a Book Exhibition in Bangalore
Palace Grounds in October 2011, when he came with certain
staff of the Institute to select books for addition to the
Institute library. Then he never forgot to ask me "Any thing
New?". I felt really glad to hand over to him the rudiments of
my new finding "Every number above 23 is a sum of powers"
and sought his blessings. This now appears in enhanced form
as WON 179. I never thought that it will be his last physical
blessings. But now I earnestly carry his heavenly blessings for
the rest of my life.

Following is the gist of a typical VKD's Maths session, which
I had the rare opportunity to attend. It was full of
information and astonishment:

To fill four blank spaces in the series 1, 7, 3, 4, 5, 1,  ,  ,  ,  ,
10, 2, - - - - - - is a task assigned by VKD, which boggled the
young brains.

What is Ramanujan's Number. Answer to this would come
out instantaneously from most of those present.

1729 = 13 + 123 = 93 + 103, sum of two different
set of cubes.

Which is the next number having similar features?

It is 4104 = 23 + 163 = 93 + 153 was my ready answer.

VKD announced, next such numbers are

64232 = 173 + 393 = 263 + 363
842751 = 233 + 943 = 633 + 843

Finally he concluded saying the sum of six different
5th powers is a perfect 5th power!

55 + 105 + 115 + 165 + 195 + 295 = X5



















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